Question: Solve for $x$ : $ 7|x - 5| - 3 = -3|x - 5| + 4 $
Solution: Add $ {3|x - 5|} $ to both sides: $ \begin{eqnarray} 7|x - 5| - 3 &=& -3|x - 5| + 4 \\ \\ { + 3|x - 5|} && { + 3|x - 5|} \\ \\ 10|x - 5| - 3 &=& 4 \end{eqnarray} $ Add ${3}$ to both sides: $ \begin{eqnarray} 10|x - 5| - 3 &=& 4 \\ \\ { + 3} &=& { + 3} \\ \\ 10|x - 5| &=& 7 \end{eqnarray} $ Divide both sides by ${10}$ $ \dfrac{10|x - 5|} {{10}} = \dfrac{7} {{10}} $ Simplify: $ |x - 5| = \dfrac{7}{10}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 5 = -\dfrac{7}{10} $ or $ x - 5 = \dfrac{7}{10} $ Solve for the solution where $x - 5$ is negative: $ x - 5 = -\dfrac{7}{10} $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& -\dfrac{7}{10} \\ \\ {+ 5} && {+ 5} \\ \\ x &=& -\dfrac{7}{10} + 5 \end{eqnarray} $ Change the ${ + 5}$ to an equivalent fraction with a denominator of $10$ $ x = - \dfrac{7}{10} {+ \dfrac{50}{10}} $ $ x = \dfrac{43}{10} $ Then calculate the solution where $x - 5$ is positive: $ x - 5 = \dfrac{7}{10} $ Add ${5}$ to both sides: $ \begin{eqnarray} x - 5 &=& \dfrac{7}{10} \\ \\ {+ 5} && {+ 5} \\ \\ x &=& \dfrac{7}{10} + 5 \end{eqnarray} $ Change the ${ + 5}$ to an equivalent fraction with a denominator of $10$ $ x = \dfrac{7}{10} {+ \dfrac{50}{10}} $ $ x = \dfrac{57}{10} $ Thus, the correct answer is $x = \dfrac{43}{10} $ or $x = \dfrac{57}{10} $.